The equation of a transverse wave traveling along a very long string is given by y = 5.2 sin(0.013π|x + 4.7π|t), where x and y are expressed in centimeters and t is in seconds.
(a) Determine the amplitude.
(b) Determine the wavelength.
(c) Determine the frequency.
(d) Determine the speed.
(e) Determine the direction of propagation of the wave.
(f) Determine the maximum transverse speed of a particle in the string.
typo I think.
not
sin(0.013π|x + 4.7π|t)
Please be more careful and recheck what you submit. I suspect a typo in that one that had (3 seconds - 1)t
To solve this problem, we will analyze the given equation step by step.
(a) The amplitude of a wave corresponds to the maximum displacement of particles from their equilibrium position. In the given equation, the coefficient of sin is 5.2, so the amplitude is 5.2 cm.
(b) The wavelength of a wave represents the distance between two consecutive points with the same phase. In the given equation, the coefficient of x is 0.013π. The general form of the wave equation is y = A sin(kx + ωt + φ), where k is the wave number, given by k = 2π/λ, where λ is the wavelength. Comparing coefficients, we have:
0.013π = 2π/λ
Simplifying, we find:
λ = 2/0.013 = 153.846 cm
Therefore, the wavelength is 153.846 cm.
(c) The frequency of a wave measures the number of waves passing a point in a given time period. In the given equation, the coefficient of t is 4.7π. The general form of the wave equation is y = A sin(kx + ωt + φ), where ω is the angular frequency, given by ω = 2πf, where f is the frequency. Comparing coefficients, we have:
4.7π = 2πf
Simplifying, we find:
f = 4.7/2 = 2.35 Hz
Therefore, the frequency is 2.35 Hz.
(d) The speed of a wave is given by v = λf, where v is the velocity, λ is the wavelength, and f is the frequency. Substituting the values we have:
v = 153.846 cm * 2.35 Hz = 361.5396 cm/s
Therefore, the speed of the wave is approximately 361.54 cm/s.
(e) The direction of propagation of the wave can be determined by analyzing the sign of the coefficient of t in the equation. In the given equation, the coefficient is positive, which indicates that the wave is propagating in the positive direction along the x-axis.
(f) The maximum transverse speed of a particle in the string is given by the derivative of the equation with respect to time. Taking the derivative, we have:
dy/dt = 5.2 * cos(0.013π|x + 4.7π|t) * 4.7π
To find the maximum value of cos, we set the argument inside the cosine function equal to either 0 or a multiple of 2π. In this case, the argument can't be zero since it's multiplied by π, so we set the argument equal to a multiple of 2π:
0.013π|x + 4.7π|t = 2π
Simplifying, we find:
|x + 4.7π|t = 153.846
Since x and t are both positive quantities, we can drop the absolute value sign. Therefore:
x + 4.7πt = 153.846
Solving for x, we have:
x = 153.846 - 4.7πt
Substituting this value into the derivative equation, we get:
dy/dt = 5.2 * cos(0.013π(153.846 - 4.7πt) + 4.7πt) * 4.7π
To find the maximum value of cos, we can evaluate this expression at t = 0 (initial value) and t = infinity (final value) since the cosine function oscillates between -1 and 1. At t = 0, we have:
dy/dt = 5.2 * cos(0.013π(153.846 - 4.7π*0) + 4.7π*0) * 4.7π
Simplifying, we find:
dy/dt = 5.2 * cos(2π) * 4.7π
At t = 0, the cosine function is equal to 1, so we have:
dy/dt = 5.2 * 1 * 4.7π = 24.74π cm/s
Therefore, the maximum transverse speed of a particle in the string is approximately 24.74π cm/s.